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We provide a new gossip algorithm to investigate the problem of opinion consensus with the time-varying influence factors and weakly connected graph among multiple agents. What is more, we discuss not only the effect of the time-varying factors and the randomized topological structure but also the spread of misinformation and communication constrains described by probabilistic quantized communication in the social network. Under the underlying weakly connected graph, we first denote that all opinion states converge to a stochastic consensus almost surely; that is, our algorithm indeed achieves the consensus with probability one. Furthermore, our results show that the mean of all the opinion states converges to the average of the initial states when time-varying influence factors satisfy some conditions. Finally, we give a result about the square mean error between the dynamic opinion states and the benchmark without quantized communication.

Individuals form beliefs on various economic, political, and social state based on information they receive from others, including friends, neighbors, and coworkers as well as local leaders and news sources. The society may face the tradeoff whether this process of information aggregation will lead to the formation of more accurate beliefs or to certain bias, which is led by the limit of communication and the change of mutual influence. Gossips, rumors, and other misinformation are an important form of social communications, and their spreading plays a significant role in human affairs. The spread of them can shape the public opinion in a country, greatly impact financial markets, and cause panic in a society during wars and epidemics outbreaks. The information content of rumors can range from simple gossip to advanced propaganda and marketing material. In practice, social groups are often swayed by misleading ads, media outlets, and political leaders, so they may hold on to incorrect and inaccurate beliefs. A central question we are interested in is to study the conditions under which exchange of information will lead to aggregation of dispersed information. We also pay attention to the gap between this consensus and the true value of the underlying state in society.

Social networks constitute a new method of studying social mechanism that has grown tremendously over the last decade. Decentralized is the inevitable trend of the development of network technology. In addition, the unprecedented number of interacting agents, the time-varying topology of agent interactions, and the unreliability of agents are key challenges for the analysis and design of this mechanism. Gossip algorithms, as an asynchronous time algorithm, have the special feature that each agent exchanges information and decisions with at most one neighboring agent in each time slot. So, gossip algorithms have been proven to be an efficacious approach to achieve fault-tolerant information dissemination. Furthermore, these algorithms can be applied in such a decentralized, large scale, and dynamically distributed network very well. In social networks, gossip algorithms to solve consensus problems have attracted a lot of interest. Based on probabilistic quantized communication, whether a group of agents has to agree under the weakly connected graph and time-varying influence factors in the communication process, starting from different initial estimates is the problem we need to study in this paper.

Consensus problems have been discussed through the great number of different opinion formation models by a lot of people. To relax the requirement of the global clock synchronization, Boyd et al. [

Since communication constraints play a major role in consensus and related problems, Carli et al. in [

We can know that the consensus problems in gossip algorithms [

Our work has been influenced by reading the papers [

In this paper, we provide a new gossip consensus algorithm based on weakly connected graph to describe the randomized agent interactions and contain probabilistic quantized communication with time-varying influence factors. The paper is organized as follows: Section

In the following, we describe briefly the distributed average consensus problem along with the proposed consensus algorithm.

We consider a set

We use an asynchronous time algorithm introduced in [

In this algorithm, without loss of generality, at most, one node is meeting another at a given time [

Independent of time and agent state, at time

an agent

for all

agent

In the social network, we want to reflect communication constraints by means of probabilistic quantization

The probabilistic quantization has been introduced in [

The following lemma gives two important properties of the probabilistic quantization.

For every

We consider a social network with the finite set

The underlying graph

Note that the graph

Here, we can have a weaker assumption and a more extensive network. In the standing assumption of [

In society, we can usually listen to advice from other people, receive the influence of others, and eventually form their own views. Due to the change in the relationship over time and the limit in the communication, we will construct our average gossip algorithm based on quantized communication and time-varying impact factors. The gossip algorithms, as the name suggests, are built upon a gossip or rumor style unreliable, asynchronous information exchange protocol. At the same time, we use the symmetric gossip algorithm which is based on mutual trust between information exchangers. Let

In the social network, we consider that probabilistic quantization

First, we know the convergence to a consensus means that a final unanimous consensus will be reached in some way. But what does the word “consensus” mean? From the view of the opinion algorithm, it means that an opinion vector in which all elements are the same. In other words, all individuals have the same opinion, which means a unanimous one. While final compromise means a compromise is reached for

Then, in order to be convenient, we will follow the assumptions as above and give the results about the consensus.

In this section, we provide our main convergence result based on the above algorithm. Particularly, we denote that despite the presence of quantized communication, with potentially very different initial opinions, the group will converge to a consensus almost surely, which all agents have the same opinion states. This consensus value is a random variable. We also provide the characterization of the expected value of this consensus under some conditions. In addition, we give a result about the square mean error between the dynamic opinion and the average initial states.

Here, we give a convergence theorem based on the topology of the underlying social network.

A global gossip consensus achieves the probabilistic consensus; that is,

This result implies that the society will reach a dynamic consensus almost surely despite the presence of the quantized communication and the effect of influence factor under a weaker assumption that the underlying graph is weakly connected. Based on the network and the pattern of communication, all agents endowed with the different initial opinion will form the common opinion with probability 1, and the expected value of the common opinion will tend to be the true value of the underlying aggregation opinion when

The following proposition provides the expectation of the error between the opinion states and the static average consensus.

The evolution of the square mean error from consensus of the algorithm satisfies

We try to study the character of its square mean convergence and find that it does not meet this convergence. But from the above proposition, we can see the square mean error has an upper bound and estimate the convergence speed of the upper bound. The limit of the bound is

Define the expected value of

From the fact that

It is easy to see that the matrix

From the above equation,

Then, the spectrum of

Define

Proof of Theorem

Define

In the second part, we will consider the character of mean about the state value.

We can use the Lebesgue dominated convergence theorem [

Then, we can derive the expression of

We can get

Noting the property of the probabilistic quantization; that is,

Therefore,

Define

Therefore,

Note that

Then, we assume

It is impossible. So,

Now we conclude that for all

Then,

Therefore, based on Lemma

So, we can get

The proof is finished.

Proof of Theorem

So,

Using the Proposition 3.4 in [

The second equality follows from the fact that

Then, repeatedly conditioning and using the iteration obtained above, we obtain, define

Define

Because of

By Lemma

In this paper, we have considered the consensus problem of gossip algorithm based on time-varying influence and weakly connected graph in the social network. Based on the gossip algorithm, we also pay attention to studying the effect of the probabilistic quantized communication.

We show that the group will achieve the probabilistic consensus value which is a random variable despite the presence of quantized communication, with potentially very different initial opinions. And we present the condition on the time-varying influence factors that guarantee the mean of consensus equals to the average initial states. We also provide a result about the square mean error which has an upper bound and the convergence speed of the upper bound can be estimated. The limit of the bound is dependent on the quantized revolution, the second smallest eigenvalue of Laplacian matrix, and the time-varying factors.

And some other interesting problems we will be concerned with in further research, such as the existence of agents who have different prejudices and whether the consensus can be reached with partial trust.

This research is supported by the National Science Foundation of China (NSFC) Grant no. 71171045 and funded by the Deanship of Scientific Research (DSR), King Abdulaziz University, under Grant no. (3-130/1434/HiCi). The authors acknowledge the technical and financial support of KAU.